Some results concerned with Hankel determinant for $\mathcal{N}\left( \mathcal{\alpha }\right)$ class
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In this paper, we give some results an upper bound of Hankel determinant of $H_{2}(1)$ for the classes of $\mathcal{N}\left( \mathcal{\alpha }\right) $. We get a sharp upper bound for $H_{2}(1)=c_{3}-c_{2}^{2}$ for $\mathcal{N}\left( \mathcal{\alpha }\right) $ by adding $z_{1},z_{2},\ldots,z_{n}$ zeros of $f(z)$ which are different than zero. Moreover, in a class of analytic functions on the unit disc, assuming the existence of angular limit on the boundary point, the estimations below of the modulus of angular derivative have been obtained. Finally, the sharpness of the inequalities obtained in the presented theorems are proved.
Abstract
In this paper, we give some results an upper bound of Hankel determinant of $H_{2}(1)$ for the classes of $\mathcal{N}\left( \mathcal{\alpha }\right) $. We get a sharp upper bound for $H_{2}(1)=c_{3}-c_{2}^{2}$ for $\mathcal{N}\left( \mathcal{\alpha }\right) $ by adding $z_{1},z_{2},\ldots,z_{n}$ zeros of $f(z)$ which are different than zero. Moreover, in a class of analytic functions on the unit disc, assuming the existence of angular limit on the boundary point, the estimations below of the modulus of angular derivative have been obtained. Finally, the sharpness of the inequalities obtained in the presented theorems are proved.
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